The proofs for Theorem REF  and Corollary REF 
are provided in Appendix . The proof techniques
exploits the convexity of \(F\)  and the Lipschitz continuity of \(\nabla F\) 
to bound \(F(w(t+1))-F(w^{*})\) . We define the error term \(e(t)=\nabla F(w(t))-g(t)\) 
and use dual averaging and some results from [1]}
to further bound \(\mbox{$F(w(t+1))$}-\mbox{$F(w^{*})$}\) .
The choice of a nonincreasing sequence of the learning rates \(\big \lbrace \alpha (t)\big \rbrace \) 
for \(t\in [T]\)  helps achieve the optimal regret and optimality gap
bounds.
